What the differences?

a) Volume increases as the cube of the length.
b) Volume increase in proportion to the cube of the length.
c) The strength of a leg bone depends on the area of its cross section, but the body it must support increases in weight by the cube of its length.

Q1. Is "as" in a) exactly the same as "in proportion to" in b)?
Q2. Or is there any difference?
Q3. What does "by" in c) mean?

I need your help.
My problem might be that I dont "get" the concepts, or the language is being used in a technical / specialist way

as an outsider to this field:

a) sounds like an incomplete sentence - so it is hard to deduce exactly what it means!
I think it needs another verb to say what the "cube of the length" is doing - or are you using "cube" as a verb?

b) makes sense in lay-person terms, though it should be "volume increaseS in proportion.."

c) seems to be trying to deal with too many ideas at once.. did you find this in a text book - it is hard to unravel as a non-specialist
I hope this will help a bit.

a) Volume increases as the cube of the length while surface area increases only as the square.

b) Yes, increases, it should have been.

c) Yes, it IS in a textbook.

How would you paraphrase the original sentence a), not using the word "as"?
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you need a mathmetician, they probably just write it this way, if that is what it says in the text book!
Okay. I'll make it very simple for you.

a) Volume increases as the cube of the length.

Is this the same as b)?

b) Volume increases as the cube of the length increases.
I still don't really know !!!! I do not think they are the same,

thinking about understanding the concept:
I think the concept is - if the length increases then the volume increases, and you can work out what the new volume will be by cubing the length,

so presumably in B) theuse of "as" isn't quite right .... you need to say: the volume increases by the cube of the length,
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How about this?

Volume increases at the same rate as the cube of the length increases.

I'm sure this does make sense. Right, Suzi?